Lyapunov Families Of Periodic Orbits Research Articles

Orbital element distribution of invariant manifolds associated with Lyapunov family of periodic orbits around L1 and L2

Abstract This study explores the initial configurations that lead to an eventual approach to a given planet, particularly Jupiter, using the invariant manifold of Lyapunov orbits around Lagrangian points L1 or L2. Reachability to the vicinity of planets would provide information on developing a process for capturing irregular satellites, which are small bodies orbiting around a giant planet with a high eccentricity that are considered to have been captured by the mother planet, rather than formed in situ. A region several times the Hill radius is often used for determining reachability, combined with checking the velocity of the planetesimal with respect to the mother planet. This kind of direct computation is inapplicable when the Hill sphere is widely open and its boundary cannot be clearly recognized. Here, we thus employ Lyapunov periodic orbits (LOs) as a formal definition of the vicinity of Jupiter and numerically track the orbital distribution of the invariant manifold of an LO while varying the Jacobi constant, CJ. Numerical tracking of the manifold is carried out directly via repeated Poincaré mapping of points initially allocated densely on a fragment of the manifold near the fixed points, with the assistance of multi-precision arithmetic using the Multiple Precision Floating-Point Reliable Library. The numerical computations show that the distribution of the semi-major axis of points on the manifolds is quite heavily tailed and that its median spans roughly 1–2 times the Jovian orbital radius. The invariant manifold provides a distribution profile of a that is similar to that obtained using a direct method.

Choreographies in the n-vortex Problem

We consider the equations of motion of $n$ vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, corresponds to choreographies of the $n$ vortices. We include numerical results for all cases, for various values of $n$, and we provide key details on the computational approach.

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of $n$ bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases $n=4,~6,~7,~8$, and $9$. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link: http://mym.iimas.unam.mx/renato/choreographies/index.html

Lissajous and Halo Orbits in the Restricted Three-Body Problem

We study the dynamics near the collinear Lagrangian points of the spatial, circular, restricted three-body problem. Following a standard procedure, we reduce the system to the center manifold and we analyze the Lissajous orbits as well as the halo orbits, the latter ones arising from bifurcations of the planar Lyapunov family of periodic orbits. To obtain the Lissajous orbits, we perform a classical perturbation theory and we provide a formal approximate solution under suitable non-degeneracy and non-resonance conditions. As for the halo orbits, we construct a normal form adapted to the synchronous resonance: introducing a detuning, measuring the displacement from the resonance, and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place. Except for a particular case, the analytical values obtained after a second order resonant perturbation theory are in very good agreement (in some cases up to the fourth decimal digit) with the numerical values found in the literature.

DYNAMICS AND BIFURCATIONS OF A NONHOLONOMIC HEISENBERG SYSTEM

We analyze both theoretically and numerically the dynamical behavior of a modification of a nonholonomic Hamiltonian system known as the Heisenberg system. The equations of motion are derived by computing the Lagrange multiplier in terms of the Poisson commutator. The presence of the constraint induces a nontrivial dynamical behavior that has been investigated by plotting the Poincaré section for the reduced system and taking advantage of a conserved quantity and the reversibilities. The dynamics is organized around two Lyapunov families of periodic orbits whose bifurcation behavior has been analyzed with a continuation technique both on the conserved quantity and on the parameters of the problem. The nongeneric branching behavior of the normal modes is theoretically explained by studying the variational equations that reduces to a Hill equation and its well known coexistence property. Invariant tori around the elliptic periodic orbits have been numerically detected but not further analyzed.

Two Classes of Cycler Trajectories in the Earth-Moon System

Anticipating the need for infrastructure in cislunar space to support telecommunications, navigation, and crew and cargo transportation, the available periodic orbits that encounter both the Earth and the moon, termed cyclers, should be characterized and their utility should be evaluated using appropriate metrics. Using the planar circular restricted three-body problem to model spacecraft motion in the Earth-moon system, a classification scheme based on resonant orbits and homoclinic connections is proposed to summarize two broad cycler classes. One class consists of high-energy near-elliptical cyclers; the second class consists of low-energy cyclers that use the dynamical structure associated with the collinear libration point between the Earth and moon. The first class is organized around resonant cyclers that can be easily characterized and are representative of the whole class. The second class of cyclers is organized by the homoclinic connections associated with the unstable Lyapunov family of periodic orbits around the libration point between the Earth and moon. The homoclinic connections bound sets of cyclers, provide close approximations of the cyclers, and can thus be used to identify their characteristics. Computational methods for both cycler classes, based on continuation and differential correction, are discussed and demonstrated.

Numerical continuation of families of homoclinic connections of periodic orbits in the RTBP

The goal of this paper is the numerical computation and continuation of homoclinic connections of the Lyapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L1, L2 and L3, of the planar circular restricted three-body problem (RTBP). We describe the method used that allows us to follow individual families of homoclinic connections by numerical continuation of a system of (nonlinear) equations that has as unknowns the initial condition of the p.o., the linear approximation of its stable and unstable manifolds and a point in a given Poincaré section in which the unstable and stable manifolds match. For the L3 case, some comments are made on the geometry of the manifold tubes and the possibility of obtaining trajectories with prescribed itineraries.

Degenerate resonances and branching of periodic orbits

In this paper we establish results on the existence of Lyapunov families of periodic orbits of reversible systems in \(\mathbb{R}^6\) around an equilibrium that presents a 0:1:1-resonance. The main proofs are based on a combined use of normal form theory, Lyapunov–Schmidt reduction and elements of symbolic computation.

Muonic helium atom as a classical three-body problem

We study the classical problem of the muonic helium atom, a helium atom with one of its electrons replaced by a muon. First, we establish the connection of the model with the one-dimensional frozen planetary approximation of the helium atom and find that there is classically stable motion for the configuration alpha-particle-electron-muon, and no stable motion for the alpha-particle-muon-electron configuration. After that, we introduce the restricted muonic helium problem, a model for the movement of an electron in the potential of the alpha particle/muon pair moving in a circular orbit. In this model, the equilibrium points, their associated Lyapunov families of periodic orbits, and their stability parameters were studied. The most interesting feature is the stability of the halo orbits, for a range of energy values. The vicinity of the alpha particle is also studied, examining Poincare sections for increasing energy values showing an increase of the chaotic motion.

The construction of periodic orbits close to triangular libration points for the three-body problem in a resistive medium

A circular, restricted three-body problem is considered when a passively gravitating point also experiences the action of small resistive forces, acting in the opposite direction to the absolute velocity vector. The nature of the loss of stability of the triangular libration points is studied using the Poincaré method in whcih the ratio of the magnitude of the resistive force to the force of gravitational attraction serves as the small parameter. Asymptotically stable periodic orbits are constructed. Two Lyapunov families of periodic orbits, which exists in the neighbourhood of the libration points of the classical theory, are the generating families. Calculations were carried out for mass ratios corresponding to the Earth-Jupiter and Earth-Moon systems, with different values of the parameters characterizing the law of resistance.

On the vertical families of two-dimensional tori near the triangular points of the bicircular problem

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and z) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, z) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system.

Bridges between the Generalized Sitnikov Family and the Lyapunov Family of Periodic Orbits

The linearization of the spatial restricted three–body problem at the collinear equilibrium point L2has two pairs of pure imaginary eigenvalues and one pair of real eigenvalues so the center manifold is four dimensional. By the classical Lyapunov center theorem there are two families of periodic solutions emanating from this equilibrium point. Using normal form techniques we investigate the existence of bridges of periodic solutions connecting these two Lyapunov families. A bridge is a third family of periodic solutions which bifurcates from both the Lyapunov families. We show that for the mass ratio parameterμnear 1/2 and near 0 there are many bridges of periodic solutions.

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2. A Lyapunov family of periodic orbits near L 2, the so-called family ( c), is born for C = C 2 and exists for values of C less than C 2. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.